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LIBRARY
Michigan State
University

 

 

 

This is to certify that the

thesis entitled

NONLINEAR DYNAMICS OF LOW IMMERSION MACHINING

presented by

Ming Liao

has been accepted towards fulfillment
of the requirements for

MS . Mednnicnl Engineering
.degree 1n

 

 

I
Major professor

 

Date 5/23/00

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NONLINEAR DYNAIVIICS OF LOW IMMERSION MACHINING

By

Ming Liao

A THESIS

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the Degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2000

ABSTRACT
NONLINEAR DYNAMICS OF LOW IMMERSION MACHINING
By

Ming Liao

A nonlinear theory of chatter in interrupted machining is developed. The machine
tool structure is represented by an equivalent single degree of freedom system with
nonlinear cutting force which obeys a digressive function (a) of chip thickness. For
highly interrupted machining processes, the system is simplified to an impact model with
the nonlinear time delay term. An approximate analytic solution of the system is obtained
from this model and the stability charts of the machining operations are estimated
analytically. Unlike the continuous cutting, the number of stability lobes is doubled. For
certain cutting conditions, the Hopf bifurcation and period-doubling bifurcation occurs
alternatively with the increase of spindle speed. Both Hopf bifurcations and period-
doubling bifurcations are subcritical as 0<0t<1. An additional nonlinearity is introduced
by the vibrating tool leaving the workpiece frequently after the bifurcation occurs. The
global bifurcations are predicted and verified by numerical simulations. Although the
results are demonstrated using an interrupted turning example, it is anticipated that the
theory will be most useful for high-speed, finish-milling operations where the radial-

depth—of—cut is only a small fraction of the tool diameter.

To my family

iii

ACKNOWLEDGEMENTS

I would like to express my deepest appreciation to the following people and
institutes for having made the completion of this thesis possible.

To the National Science Foundation for funding some related research which let
to the research in this thesis (NSF, grant No. DMI-9800323).

To my advisor Dr. Brian Feeny, for his guidance and invaluable assistance,
without which this thesis would not have been possible.

To Dr. Matt Davis, National Institute of Standards and Technology (NIST), on the
basis of whose work this paper was begun to be developed in the summer of 1999. His
dedications to the subject and personal integrity have served as an example and an
inspiration to me.

To my other committee members, Dr. Steve Shaw and Dr. Dinesh Balabangadhar,
for their kind help and useful advice.

To Dr. Jon Pratt, and Dr. Tim Burns, NTST, for the fruitful discussions we had.

To all my other instructors and colleagues at MSU, for their skillful instructions in
the classroom and lab, but more importantly, for their friendship and constant
encouragement.

To my family, for years of support, patience and understanding.

iv

TABLE OF CONTENTS

LIST OF FIGURES .................................................................................. vi

KEY TO SYMBOLS OR ABBREVIATIONS ................................................. viii
CHAPTER 1

INTRODUCTION .................................................................................... 1

1.1 Machine Dynamics ..................................................................... 1

1.2 Low Immersion Milling ................................................................ 8

1.3 Nonlinear Dynamics of Machining ................................................. 12

1.6 Contributions .......................................................................... 16
CHAPTER 2

THEORETICAL ANALYSIS ..................................................................... 18

2.1 Modeling of the Interrupt Cutting ................................................... 18

2.1.1 Basic Assumptions ......................................................... 18

2.1.2 Modeling of Interrupted Turning ....................................... 21

2.1.3 Modeling of Low Immersion Milling .................................. 24

2.2 Approximate Analytic Solution ..................................................... 24

2.3 Local Stability ......................................................................... 28

2.4' Local Bifurcations ...................................................................... 38

2.5 The Role of 0c ........................................................................... 46

2.6 Post Bifurcation Orbits ............................................................... 53
CHAPTER 3

SIMULATIONS AND EXPERIMENTS ........................................................ 66

3.1 Simulations ............................................................................ 66

3.2 Experiments ............................................................................ 75
CHAPTER 4

CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ............................ 77

BIBLIOGRAPHY ................................................................................... 80

1.1

1.2

1.3

1.4

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

LIST OF FIGURES

Regeneration in turning ..................................................................... 2
Stability chart for a turning operation .................................................... 6
Sketch of Low Immersion Milling ........................................................ 9
Stability curve for uninterrupted and highly interrupted turning operations ....... 11

Schematic of an interrupted turning operation .......................................... 19
Schematic of a low immersion milling operation ...................................... 19
The relation between cutting force and chip thickness ................................ 21
Cutting force series in interrupted machining .......................................... 27
The ways the eigenvalues travel through the unit circle .............................. 33
Stability lobes diagram of interrupted turning .......................................... 35
Parameter traces for bifurcations ......................................................... 39
Subcritical period-doubling bifurcation diagram ....................................... 42
Subcritical Hopf bifurcation diagram ................................................... 46

The comparison of stability charts from the nonlinear model and the linear

model ......................................................................................... 48
Period-doubling bifurcation diagrams at different values of or ....................... 50
Hopf bifurcation diagrams at different values of or .................................... 52
Hopf bifurcation diagram in continuous cutting ........................................ 55
Global bifurcation diagram for period-doubling bifurcation case ................... 59

vi

2.15 Possible ways that chatter amplitude stabilizes at the cut-no-cut period-two orbit

................................................................................................ 60
2.16 Global bifurcation diagram for Hopf bifurcation case ................................. 63
2.17 Possible ways that chatter amplitude becomes quasiperiodic ........................ 64
3.1 Comparison of stability lobes obtained from analytic solutions and from

simulations .................................................................................. 69
3.2 Stable period-two orbit after the subcitical period-doubling bifurcation ........... 71
3.3 Quasiperiodic orbit after the subcitical Hopf bifurcation. ............................ 71
3.4 Comparison of in-cut passes and out-of-cut passes for quasiperiodic orbits. .......72
3.5 Simulations of the subcritical period-doubling bifurcation diagram. . . . . . . . . . .......74
3.6 Simulations of the subcritical Hopf bifurcation diagram ............................. 74
3.7 Comparison of stability charts from experiments of 5% immersion and 25%

immersion end-milling .................................................................... 76
3.8 Comparison of stability charts from analytical prediction and experiments

of 5% immersion end-milling ............................................................. 76

vii

SYMBOL

f0

(on
ma

fa
f4

'm

KEY TO SYMBOLS OR ABBREVIATIONS

MEANING UNITS
Equivalent mass of the system kg
Damping coefficient of equivalent dashpot N.s/m
Stiffness coefficient of equivalent spring N/m
Thrust force coefficient N/m2
Axial depth of chip m
Radial depth of chip m
Tool engagement period s
Spindle speed 1/s
transition time for free vibrations s

The number of flutes on the end-mill
Feed rate m/flute or m/rev

Damping ratio

Natural frequency of the system rad/s
Damped frequency of the system rad/s
Natural frequency of the system Hz
Damped frequency of the system Hz

immersion ratio
Exponent of digressive force function

Limit chip width In

viii

CHAPTER 1

INTRODUCTION

Section 1.1: Machine Dynamics

In machine dynamics, one big issue is the conditions of the occurrence of the self-
excited vibrations called “chatter”.

Chatter is caused mainly by “regeneration of waviness” on the machined surface.
Regeneration is possible because in almost all machining operations the tool removes the
chip from a surface which was produced by the tool in the preceding pass, i.e., the surface
produced in tuning during the preceding revolution or, in milling, by the preceding tooth
of the cutter.

For example, in turning operations, if there is relative vibration between tool and
workpiece, waviness is generated on the cut surface. The tool in the next revolution
encounters a wavy surface and removes a chip with periodically variable thickness. The
cutting force is then periodically variable. This produces vibrations, and depending on
conditions derived later, these vibrations may be at least as large as in the preceding
revolution. The newly created surface is again wavy, and in this way, the waviness is
continually regenerated. In a “stable” case any initial vibration diminishes in subsequent
passes; in an “unstable” case it increases, and at the “limit of stability” the magnitude of

the vibration remains constant. Figure 1.1 shows the regeneration in turning operations.

 

Figure 1.1 regeneration in turning

Milling is a much more complex case than turning as regards chatter generation.
In milling, the “subsequent passes” are represented by the cuts of the individual teeth of
the cutter, each of them regenerating the waviness created by the preceding tooth. So the
phasing between those subsequent undulations is not determined by the number of waves
per revolution as in turning, but by the waves per tooth spacing. It means, basically, for
the same spindle speed, the number of waves between teeth may be small, or even to
zero.

When chatter occurs, the machined surface is spoiled by chatter marks. Vibrations
can be so strong that the tool may leave the workpiece, i.e., the amplitude of the vibration
between tool and workpiece becomes larger than the chip thickness. Cutting force can
oscillate with peaks of at least double the mean value. These vibrations, especially in
machining stronger materials, might lead to the breakage of tools. This aspect as well as
the aspect of surface finish requires that machining parameters be kept in a region for
which the chatter does not occur. This generally limits the material removal rate. In

aircraft manufacture, i.e., mostly in end milling of aluminum, chatter is unavoidable and

many hours of labor are spent on manually removing chatter marks which would
otherwise decrease the fatigue life of the machined parts.

Whether or not chatter occurs (unstable case) depends on the structure of the
machine and workpiece, the stiffness, damping and orientation of its modes of vibration,
and on the conditions of the cut: workpiece material, feed, cutting speed and chip width.
For a given operation, the machine tool structure and the workpiece are not freely
selectable, while the cutting parameters, like feed, speed, and chip width, are tunable for
an optimal productivity. It is recognized that the most significant cutting parameter
which is decisive for the generation of chatter is the width of cut (chip width) (Tlusty,
1993), denoted as w. For sufficiently small chip widths cutting is stable, without chatter.
By increasing chip width w, chatter starts to occur at a certain value—limit chip width,
denoted as w;,-,,,, and becomes more energetic for all values of w> wum. The value of Wu,"
depends on the dynamic characteristics of the structure, on the workpiece material, on the
cutting speed and feed, and on the geometry of the tool.

Extensive research has been devoted to the problems of chatter in machining.
Arnold (1946) first suggested regeneration of waves on the workpiece as a potential
cause of chatter. Tobias and Fishwick (1958), Tlusty and Polacek (1957, 1963) and
Merrit (1965) did notable experimental and analytical efforts. They built mathematical
models of machining and represented the stability regions in the form of stability charts.
Following these pioneering efforts, there have been many efforts to understand
regenerative stability in machining operations.

Shridar (1968) developed a mathematical model for general milling processes. In

his paper, a more complete description of the milling process is formulated. The resulting

equation is a general nth order vector-matrix linear equation with periodic coefficients
and a transport lag. Or equally, it is a set of linear first-order differential equations (state
variable form), convenient for stability analysis. He concluded that the chatter analysis
was associated with the stability characteristics of linear differential equations with
periodic coefficients and it did not appear that stability methods based on frequency
analysis can be used to study chatter in a milling operation.

Shridar (1968) also gave a stability algorithm for the general milling process. The
algorithm can be used in conjunction with the digital computer and developed as a means
of analytically determining the stability of the linear differential equation. The algorithm
permitted the dtermination of the stability boundaries in the space of controllable
parameters associated with a cutting operation.

Tlusty, Ismail, Zaton and Smith (1981, 1983, 1986,1991) studied milling stability
by means of time domain simulations. They showed that high gains of stability are
achievable by determining and using a particular spindle Speed such that the cutter tooth
frequency approaches the frequency of the decisive mode of vibrations as measured on
the cutter. Specifically, they made the following conclusions on high speed milling: the
process damping which stabilizes cutting at conventional speeds is absent in high-speed
milling; high-speed, high-power face milling and short end milling can only be stable if
the rigidity of high-speed spindles is substantially increased; that high-speed milling with
long end mills is much more difficult from the point of view of chatter because of the
inherent low stiffness of the tool itself. They also pointed out the influence the possibility

of interactions between regenerative vibrations and vibrations induced by the intermittent

engagement of the tool and workpiece. However, the possibility that the intermittency
itself could lead to vibration instability was not discussed.

Altintas and Budak (1995) produced analytical stability lobes in the milling
operation. The stability model required transfer functions of the structure at the cutter-
workpiece contact zone, static cutting force coefficients, radial immersion and the
number of teeth on the cutter. Time varying dynamic cutting force coefficients were
approximated by their Fourier series components, and the chatter free axial depth of cuts
and spindle speeds were calculated directly from the proposed set of linear analytic
expressions without any digital iteration. The result was in excellent agreement with the
time domain simulations and multi-frequency solutions

Recently, there have been more researchers working on the problem of chatter.
Many methods have been proposed to prevent chatter in machining. Typically, three
approaches are preferred. The first is to design and set up the right kind of machine tool
structure which leads to improved stability. The second is to choose optimal spindle
speeds, and the third is to design and manufacture milling cutters with non-uniform tooth
pitch and with alternating helix as well as with serrated or undulated edges.

In a machining system, the machine tool structure and tool geometry are already
given. It is more feasible to choose the optimal speeds to increase the stability and
productivity. Fortunately, at the limit of stability, the width of cut and the spindle speed
have a relationship such as that shown in the stability chart of Figure1.2. The stability
lobes shown here mark the boundary between stable and unstable cuts of operation, the
area below the lobes representing stable operation zones and the area above, unstable

operating zones. It can be seen from the figure that at some optimal speeds, the limit chip

width can be much larger than the others, and the higher the spindle speed, the larger the
limit chip width. So, by increasing the spindle speed, not only the productivity is
increased, but the stability is also improved. This is what makes the high-speed

machining more and more prosperous.

A

Chip width

 

 

 

 

Spindle Speed

Figure 1.2 Stability chart for a continuous cutting operation ( above
the stability lobes are chatter regions; below are stable regions.)

Most recently, high-speed machining technology has been rapidly
commercialized in industry. Researchers found that the high-speed machining has never
been so important in industries until today. In less than 20 years, the spindle speeds, feed
rates and power capabilities of production-grade machining centers have increased by
more than a factor of ten. Currently, commercially available machining centers can be
obtained with spindles capable of speeds exceeding 40 thousand revolutions per minute
(krpm) while delivering 40 kW of continuous power to the cutting zone, and slides
capable of producing linear motions at speeds and accelerations exceeding 1 m/s and 10

m/32, respectively (Davies et al., 1999a).

In the aerospace and automotive industries, the high-speed machining technology
has been applied widely, especially in manufacture of aluminum components where
volumetric material removal rates can be extremely high, often approaching thousands of
cubic centimeters per minute. The advantages of high-speed machining include:

(1) shorter machining time;

(2) improved surface finish;

(3) reduced thermal and mechanical stresses on the workpiece and tool; and

(4) improved dynamic stability.

Halley and Helvey (1999) at Boeing, and Tlusty, Smith, and Ismail (1991, 1996)
at University of Florida, did a lot of research on high-speed machining on the
manufacture of aluminum components, the results are very encouraging. But progress in
high-speed machining of other materials has been more limited except some special
examples.

Of many factors which affect the material removal rate in high-speed machining,
two primary limitations are:

(1) tool wear; and

(2) dynamical stability.

In high-speed machining aluminum components, with tungsten carbide cutting
tool, the tool wear rates can be ignored. The primary limitation is dynamic stability.
Fortunately, as predicted in Figure 1.2, in the high-speed region, the chatter lobes are
much less densely packed, and the limit chip width is dramatically increased at the
optimal spindle speeds. This means increasing the spindle speed can improve the

dynamical stability.

Unfortunately, in high-speed machining more difficult materials, the tool wear
can no longer be ignored. High temperature caused by high-speed machining and high
material removal rates can lead to rapid catastrophic tool failure. To reduce the tool wear,
there is a tendency to decrease the cutting speed. For high-speed machining materials that
cause non-negligible tool wear, a compromise must be struck between reduced wear rates
and improved stability. This is not a preferred choice. One alternative strategy is to use

the low immersion machining.

Section 1.2 Low Immersion Machining

According to Davies et al. (1999a), low immersion machining means that the
radial depth-of-cut is very low so that the partial immersion ratio p (defined as the ratio
of time spent in cutting to not in cutting) is very small, sometimes as low as a few
percent. Low immersion is often used in high-speed machining, especially in high-speed
milling. In Figure 1.3, the schematics show the comparison between low immersion

machining and deep immersion machining.

 

 

 

 

 

Figure 1.3 (a) Sketch of Low Immersion Milling

 

 

 

 

Figure 1.3(b) Sketch of Deep Immersion milling

The low immersion machining has the following advantages.
(1) Reducing the tool wear. The reason for this is that during the portion of the
time when the tool is not cutting, the water (coolant) runs directly on those

portions of the lip surface and cutting edge of the too] which do the work. For

this reason the tool is more effectively cooled with intermittent work than with
steady work (Taylor, 1907).

(2) Increasing the stability. First, a rather higher cutting speed can be used with
an intermittent cut than with a steady one because of the reduced tool wear.
Second, with highly interrupted cutting, the tool has more time in damping to
overcome the oscillating energy obtained from cutting.

(3) Better surface finish. This is straightforward and is especially suitable for
sculpted surfaces.

(4) Suitable for near-net-shape workpieces. With the development of
manufacture technologies, the near-net-shape workpieces are more and more
widely used to reduce the total amount of material that must be removed by
machining.

With reduced tool wear and increased stability, the spindle speed can be very

high. The limit chip width could be significantly increased at the optimal spindle speeds.

The loss of material removal rates caused by reduced radial depths of cut can be

compensated by increased axial depths of cut. Overall this could produce substantial

improvements in the attainable material removal rates. Recently, Smith et al. (1999) did

some investigations on the effect of reduced radial immersion on high-speed machining

of titanium alloys, with encouraging results.

The major feature of low immersion machining is the highly interrupted cutting.

The dynamics of low immersion machining is dramatically different from the deep

immersion machining or continuous cutting because the intermittent cut introduces an

impact-type nonlinearity. Each of the stability lobes in continuous cutting is split into two

10

lobes because of the impact induced dynamics, as shown in Figure 1.4. The total number
of stability lobes of interrupted cutting is doubled compared to continuous cutting. The
newly found stable regions in stability chart is doubled compared to continuous cutting.

The conventional model based on the continuous cutting is no longer valid to predict the

stability for interrupted machining.

 

 

 

 

 

 

10 :r 'r E
E p=5% j
A E I" E 1‘
g 5 - g I", :1 I”,
a i I” “\ ”I,
/ p: 100%
O
I T I
25 50 75 100
Q (krpm)

Figure 1.4 Simulated stability lobes for uninterrupted and highly interrupted
turning operations. (p=5%: highly interrupted turning; p=100%: uninterrupted
turning; parameters: k=l.4MN/rn, m=0.043kg, c=8.6N-s/m) Courtesy of M. A.

Davies.

Davies et al. (1999a) developed a linear theory for the interrupted machining.
Using two examples, interrupted turning and low immersion end-milling, he built a linear
model for interrupted machining. The machine tool structure was represented with a

single-degree-of—freedom system with a linear delayed term. The system then was treated

11

as a two-stage map. For the first stage, the tool was not in contact with the workpiece.
The system was just in free vibration and the solution was straightforward. For the
second stage, the tool was engaging in cut. There was no general analytic, time-domain
solution for this stage, instead, an approximate analytic solution was sought. With the
assumption of a very small immersion ratio, the time of contact was very short compared
to the characteristic period of vibration for the system. The result of the cutting action
was to provide an impulsive change in the momentum of the oscillating body, and the
system reduced to a kicked harmonic oscillator with delay (Zaslavsky et al. (1978,
1991)). The kicked-oscillator produced a simple map. The stability was analytically
determined. This linear theory predicted the doubling in the number of stability lobes.
This had significant meaning for the determination of the optimal cutting conditions for
low-radial immersion high-speed machining.

However, because of high spindle speeds and high-slide accelerations,
maintaining accuracy in high-speed machining requires the development of more
accurate dynamic models of the machine and cutting process. Linear models are
inadequate to obtain the enough accuracy and they are also incompetent for explain many
nonlinearities in machining. Nonlinear models are often required where linear models

fail.

Section 1.3 Nonlinear Dynamics of Machining

According to the linear theory of chatter, the amplitude of vibration ought to

increase indefinitely once the width of cut exceeds a certain critical value and the process

12

becomes unstable. This is not so in practice when the amplitude, after a rapid initial
increase stabilizes itself at a finite level. This stabilization phenomenon is explained by
most investigators as being due to nonlinearities in the system.

An exploration of nonlinear region of chatter is of importance also from the point
of view of correlating experimental and theoretical results. The linear theory predicts the
threshold of stability in the form of a stability chart. In actual experiments such threshold
conditions cannot be set up and the chatter vibration actually investigated is already
stabilized by nonlinear effects. As a result, experimentally determined stability charts
may differ considerably from those predicted theoretically.

The linear theory is inadequate also for explain “finite amplitude instability”
(Tobias(l974)). Systems which show this effect have the characteristics of being stable
for small disturbances but unstable when these become sufficiently large, as may arise in
intermittent cutting.

Probably Hanna and Tobias (1974) are the earliest researchers who investigated
the nonlinearities in chatter analysis. They represented the machine-tool structure by a
single-degree—of-freedom system with nonlinear stiffness characteristics and modeled the
cutting force as a third degree polynomial of the chip thickness. This model led to a
second-order differential equation with nonlinear stiffness and nonlinear time-delay terms
that was analyzed using a two-term harmonic balance.

Shi and Tobias (1984) improved this model by showing that “finite amplitude
instability” could be understood without assuming a non-linear behavior of the machine

tool structure; it was due to the regenerative conditions arising when the tool left the

13

workpiece material and the nonlinearity of the cutting force as a function of the chip
thickness.

In contrast, Tlusty and Ismail (1982) turned to numerics to characterize the
nonlinearity associated with the tool leaving the workpiece. They developed a time-
domain simulation using a two-degree-of-freedom model of the machine-tool structure.
The cutting force is assumed to be proportional to the chip thickness when the tool is
cutting and zero whenever the tool vibrates out of the cut. In addition, the chip thickness
at any time t depends on the tool displacements at previous times t-T, t-ZT, t-3T, etc. This
dependence is referred to as “multiple regenerative” effect.

Grabec (1986, 1988) used a linear two-degree-of-freedom model from machine-
tool structure and coupled the two degrees of freedom through a friction coefficient. He
did not consider regeneration. The cutting force was modeled as a quadratic function of
the chip thickness and velocity and was derived from the experimental results reported by
Hastings et al. (1971, 1980) for stable cutting. Grabec numerically integrated the
equations of motion and found that the vibration undergoes a bifurcation from
quasiperiodic to chaotic as the mean cutting force was increased.

Lin and Weng (1990) used a single-degree-of-freedom model with regeneration
and non-linear dependence on chip thickness and velocity to investigate the nonlinear
stability limit. They used the method of multiple scales to determine analytically a limit
width of cut for orthogonal cutting. They have also considered a two-degree-of-freedom
model with the addition of multiple regeneration (Lin and Weng, 1991). This study was

numerical and demonstrated chaotic dynamics.

14

A single-degree-of—freedom modal incorporating time-delay effects and nonlinear
velocity-dependent friction force at the tool/workpiece interface was investigated by
Moon (1994). He suggested that chaos would occur in the model due to the time-delay
effects and the strong nonlinearity. However, numerical integration of the equations of
motion revealed only periodic and quasiperiodic motions, and Moon concluded that a two
degree-of-freedom model might be required to observe chaos.

Using perturbation methods and bifurcation theory, Nayfeh et al. (1997) restudied
the model built by Hanna and Tobias. Using the method of multiple scales, a normal
form of the Hopf bifurcation was obtained by including the effects of the quadratic and
cubic nonlinearities. Using a six-term harmonic balance solution, two cyclic-fold
bifurcations were found to result in large-amplitude periodic solutions, hysteresis, jumps,
and subcritical instability. As the width of cut w increased, the periodic solutions
underwent a second Hopf bifurcation, leading to a two-period quasi-periodic motion (a
two-torus). The periodic and quasiperiodic solutions were verified using numerical
simulation. As w increased further, the torus doubled. Then, the doubled torus broke
down, resulting in a chaotic motion.

However, the effect of the intermittent cut on the nonlinearity of the process has
not been adequately treated. In conventional nonlinear models of machining, the cutting
is continuous. The dynamics of system can then be described by a nonlinear differential
equations with nonlinear delayed terms. The state of the system is continuous as a
function of time. But for interrupted machining, the cutting is not continuous, and the
state of system is not continuously changing. Instead, the system is described by a two-

stage map. In conventional model of continuous cutting, only Hopf bifurcations are

15

possible. But in interrupted machining, not only Hopf bifurcations, but period-doubling
bifurcations are also possible to occur at certain conditions. Actually, with other
parameters fixed, the Hopf bifurcations and period-doubling bifurcations occur
alternatively with the increase of the spindle speed. In this paper, for the first time, both
Hopf and period-doubling bifurcations of interrupted machining are analytically studied.
In Chapter 2, we develop a nonlinear theory for the interrupted machining. Based
on the linear theory of Davies et al. (1999a) on interrupted turning and low immersion
end-milling, a nonlinear model is built. The system is represented by a single degree-of-
freedom system, too, but with a nonlinear delay term. The cutting force is a digressive
function of the chip thickness. From this model, the analytic solution of the stability is
obtained. Both Hopf and period-doubling bifurcations are found and analyzed. The

following section summarized the contributions of this thesis.

Section 1.4: Contributions

1. An nonlinear model of interrupted machining is built. The machine tool structure
is represented by an equivalent single degree of freedom system with the cutting
forces by a digressive function (or) of chip thickness.

2. For highly interrupted machining, the system is simplified as an impact model and
an analytic solution is obtained. Besides Hopf bifurcation, Period-doubling
bifurcation is also found in interrupted machining. The number of stability lobes

is doubled as in the linear results of Davies et al. (1999a).

16

3. The stability chart is analytically estimated using the nonlinear model. In the
linear model of Davies et al. (1999a), the feed rate f0 did not affect the stability of
the system, but because of the nonlinearity of the digressive cutting force, f0 plays
a role in determining the stability chart of the nonlinear model. Compared to
linear model, the stability lobes of nonlinear model are pulled down. Thus,
stability lobes of linear model are misleading, and can lead to false sense of
security.

4. Both Hopf bifurcation and period-doubling bifurcation are subcritical as 0<0t<1.
The effect of the value of or on the bifurcations is analyzed. When O<0t<1, the
bifurcation is subcritical; when 0t=1, the system becomes linear, and the chatter
amplitude would be infinite; when 0t>l, the bifurcation is supercritical.

5. The additional nonlinearity is introduced by which the vibrating tool leaves the
workpiece frequently as the bifurcation occurs. Post-bifurcation orbits are
analyzed. Period-two orbits and quasiperiodic orbits are predicted. Possible routes
to chaos are discussed.

6. Numerical simulations verify the analytic results above. Hysteresis is found for
subcritical bifurcations. The stable period-two orbit and quasiperiodic orbit are

found and well agree with the analytic solution.

17

CHAPTER 2

THEORETICAL ANALYSIS

Section 2.1: Modeling of the Interrupt Cutting

2.1.1: Basic Assumptions

To derive the theory for the stability of intermittent machining, two example

problems will be considered: turning of a workpiece with a raised area as shown in

Figure (2.1); and low-immersion end-milling with an N-fluted end-mill as shown in

Figure (2.2). In each case the spindle speed is [2, measured in revolutions per second

(RPM), and the spindle period T is defined to be the inverse of the spindle speed. 8 is the

feed rate in meters per second. For each case, the following assumptions are made.

18

 

 

 

 

 

 

 

 

 

Figure 2.2 Schematic of a low immersion milling operation.

19

1.

Machine Tool Structure. For the sake of simplicity it will be assumed that
the mode of vibration of the machine tool structure which under certain
conditions may become unstable can adequately be represented by an
equivalent single degree of freedom system, specified by an equivalent mass
m, an equivalent dashpot, and an equivalent spring.

It will further be assumed that the structure can vibrate only in the direction
normal to the machined surface and that hence the machine tool system can be
represented by models of the type shown in Figure 2.1 for turning and Figure
2.2 for end-milling.

Stiffness. According to Shi and Tobias (1984), there are two sources of
nonlinearity. Firstly, by the chatter amplitudes exceeding a certain value,
dependent on the mean chip thickness and the vibrating tool leaving the
workpiece. Secondly, by a non-linearity of the cutting force characteristics.
Machine tool structural nonlinearity has less effect on finite amplitude
instability. So here we assume that the stiffness function is linear.

Damping. In high-speed machining, machine tool structural damping is often
very small. Usually the damping ratio is a few percent, and frequently less
than one percent (Davies (1999)). So here we assume the damping function is

also linear and represented by the damping ratio ;

Cutting Force Function. The normal cutting force is assumed to be a
digressive function of the chip thickness (Stephan, (1999)), which can be

expressed as:

F = CK,da (2.1)

20

Where F is the cutting force, d is the chip thickness, K, is the generalized
thrust cutting force coefficient which is proportional to the width of cut w, and

0<a<l. C is determined by other cutting conditions. The relation between the

cutting force and the chip thickness is shown in Figure 2.3:

Feed“ll

Cutting force

 

 

Chip thickness

Figure 2.3 The relation between cutting force and chip thickness

3. Very small immersion ratio p. The time of contact is a small fraction of the
period for one engagement between the workpiece and the tool, and is very

small compared to the characteristic response time of the system.

Assumptions 1 and 2 are similar to those models built by previous researchers.

Assumption 3 is the key to the approximate analytical solution presented later. Next, we

will derive the equation of motions for turning and milling respectively.

2.1.2: Modeling of Interrupted Turning

21

A simple form of interrupted cutting operation is the turning of a small raised area
on a round workpiece as shown in figure 2.1. For simplicity, all flexibility is assumed to
be perpendicular to the workpiece surface (y-direction). The ith revolution of the

workpiece is defined to be begin at time rH 2 (i —1)T. During this revolution, cutting

begins at time t,’ = (i — ,0)T and ends at time t, = :7. Applying the assumption 1 and 2,
the equation of motion is written as:
my+cy+ky=—K,wda (2.2)

m — equivalent mass of the system;

c — damping coefficient of equivalent dashpot;

k— stiffness coefficient of equivalent spring;

K, — thrust force coefficient, where the negative sign before it is introduced to

allow for the opposite directions in which increases of y and d are measured;

w — axial depth of chip or chip width;

d — radial depth of chip or chip thickness, and it is given by the following

relations,
(I = 0, re ((i — 1)T,i(1 - p)T) (2 3)
d = r0 + yo) — ya —T>. re (ta — pm?) '
T — tool engagement period, which is the inverse of spindle speed [2 in

interrupted turning and UN!) in low immersion milling, where N is the number of
flutes on the endmill;

f0 —- feed per engagement and fo = ST;

y(t) — current displacement of the tool

22

y(t-T) — displacement of the tool during the previous engagement.

By using damping ratio { and natural frequency of the system a)", the equation

 

(2.2) becomes,
y+25wny+wny =-£’;11d“ (2.4)
where
g“ = c - a) = 1‘- (2.5)

The component of velocity in the y-direction is defined as,
V0) = 5’0) (2.6)
For the ith revolution of the workpiece, the solution of equation (2.4) proceeds as

follows. At time t,._l , the position and velocity of the system are denoted (y,._l ,vH) . This

serves as an initial condition for equation (2.4) with the chip thickness d set to zero

according to equation (2.3). The system undergoes a simple damped, harmonic oscillation
until time If. At this time, the new position and velocity are denoted (y; ,vf). This

serves as the initial condition for equation (2.4) with non-zero chip thickness. The

solution then proceeds until time t, resulting in the new position and velocity (ynvi),
which is then used as the initial condition for the next revolution (1' +1). The procedure
begins with some set of initial conditions at time t0 and proceeds ad-infinitum with

repetition of the above-mentioned steps. Thus the solution for each revolution defines a
two-stage map taking conditions (y,._l ,v,_,) to (y, ,vi). The first stage of the map has an

analytic time-domain solution. In general, the second does not.

23

2.1.3: Modeling of Low Immersion Milling

For the most simplified case, low immersion endmilling operations produce
equations whose stability can be approximately determined by studying equations (2.2)
and (2.3). To see this, consider the low immersion endmilling operation shown
schematically in Figure 2.2. Making the assumption that the horizontal and vertical (y-
direction) motions are not coupled through the machine tool structure (i.e., diagonal
mass, damping, and stiffness matrices) and applying assumption 3, the following

equations of motion for the cutter are obtained (see Davies et al. [30]):

y+2§wni+w3y=-fld“ (2'7)
m
where
d = 0, t e ((i — 1)T,i(1 — p)T) (2 8
d :3 y(t) — y(r —T), E (i(1-P)T.iT) . )

Compared to equations (2.3) and (2.4), the only difference here is that equation
(2.8) is absent a feed—rate determined driving term that does not directly affect the
stability of the system. Thus for very small p, the approximate stability behavior of low

immersion milling is the same as that for the interrupted turning.

Section 2.2: Approximate Analytic Solution

In this section, we derive an approximate time-domain solution for interrupted
cutting operations. The key to the analytic stability calculation is to find an approximate

solution for the map describing the evolution of the interrupted cutting. The map must be

24

developed in two phases: (1) the free vibration phase that maps the state (y,._l ,vH) to the
state (y,- ,v,‘ ); and (2) the cutting or delayed forcing phase that maps the state (y; ,vi’)

to the state (y, ,v, ). We begin by considering interrupted turning. Low immersion milling

can be treated in a similar manner.
For ith revolution, the free vibrations evolve according to the state-transition
matrix for a damped harmonic oscillator (see Hirsch and Smale (1974)). Given the state

at the end of cut number i—l, denoted (y,_1,v,_,), the state at the beginning of cut

number i, denoted ( yi- ,vf) is given by the following expression:

rm

where A is the state-transition matrix and can be expressed as follows:

 

 

 

(
Ca)" sin(wd 2') + cos(a)d 1') J—sin(a)dr)
A = e-W “’d _ 2 “’d {m (2.10)
" sin(a)dr) cos(a)dr)— " sin(wdz')
K 01d cod )

 

 

where

a), = wnJIT-E; (2.11)

which is the damped natural frequency for the free oscillator, and
z = (1 — p)T (2-12)
which is the transition time. This fully describes the evolution of the system form the

initial state to the state just prior to the engagement with the workpiece.

The derivation of an approximate solution for the second portion of the map

taking (y; ,v,’ ) to (y, ,v,) requires the use of assumption 3. Its validity is supported by a

25

comparison of the approximate order of magnitude of the system response time at the
onset of instability (given by the inverse of the classical chatter frequency) and the order
of magnitude of the time of contact as the fractional time-of-contact p is decreased.

The response time of the system can be estimated using the expression for the
chatter frequency for the case where ,0 =1. For 4 << 1, Tobias (1964) shows that this

frequency can be approximated by,

K
601.... fi/Z—‘WC +w§ (2.14)
m

where mum is the chatter frequency and a)

 

is the critical width-of-cut for instability in

lim

the case where ,0 =1; note that a) is a function of the spindle speed and thus the

lim
chatter frequency changes as we sweep along the traditional lobe diagram. Using

equation (2.14), the approximate response time of the system, denoted :- , is given by

E. Empirically, it has been observed that as the immersion ,0 decreases, the critical

lim
width-of-cut scales as i at all spindle speeds (Davies et al. 1999a). This observation,
,0

when combined with equation (2.14) suggests that the characteristic period of the system

1 . . . .
should scale as — for small ,0 . However, p is a linear function of the time of contact.

J5

Therefore as ,0 decreases, the time of contact decreases more rapidly than the
characteristic period of the system. This implies that for p small enough, the tool can be

assumed to remain at a fixed position during each subsequent cut. The surface location
resulting from each cut is determined by the position of the tool when the cut is

commenced. The velocity of the tool changes in response to the finite impulse it

26

experiences during the cut. Thus assumption 3 greatly simplifies the dynamics of the tool,
and allows an analytic solution to be obtained for the stability of the system.

Assume that the position of the oscillator remains constant during the cutting
interaction so that y, = y: and keep in mind that p is very small, the cutting forces are

impulses simplified as shown in Figure 2.4. So the approximate velocity of the system

after the cut can be found by using the impact model (Davies et a1. (1999)),

_ K, , a
(vi “VI. )z—WQp—W'Cfo +y; -y;_1) (2.16)

 

 

A
0 Actual Force
0
‘5
LL.
time
A
0 Simplified Force
8
r2

 

nil rill.

"H“ 0 time

Figure 2.4 Cutting force series in interrupted machining

Applying these assumptions we obtain the following expression for the second

portion of the map in matrix form,

27

y" = y: + Km 0 (217)
V, vi“ “KL/Po + yi " yin)“ .

Combining equations (2.9) and (2.17),
0
yr yi-l
: + K, a 2.18
(vi) {vi—l] [_—n_19&w_(fo+Yr-yr’-r) ] ( )

A similar treatment of low immersion milling produces a map with same stability

behavior as equation (2.18) with the relevant engagement period now being 35 instead

of g and with fo set to zero. The map thus has the form,

0
y‘ _ yi-l K
{vii—{V.-i]+[‘—£(yi “yH)a] (2.19)

This form makes physical sense since we expect that for very low immersion
milling, the periodic driving term is very small, but displacements in the sensitive
direction y still produce substantial changes in the force. Thus, both systems are modeled
by a kicked harmonic oscillator with delay. Next we will determine the stability of this
model. Since the interrupted turning is a simplified model of low immersion milling and
they are qualitatively similar in dynamics, the rest of this work focuses on the analysis of

equation (2.18).

Section 2.3: Local Stability

28

To determine the stability of the equation (2.18), the fixed point of the map is first

determined and then the stability of motions about that fixed point are examined. A fixed
point (9,5,) remains unchanged on further iteration of the map. So in equation (2.18),

let

(Wit?)
vi v,._l v
We can get
[.)=(I-A)“ _K,pw a (2.21)
v —m_§2_f°

where I is the identity matrix. The fixed point represents a forced periodic motion of the
tool with period T in turning and -1% in milling. The local stability of this periodic motion

is determined by the linearization of the map about the fixed point.
To get the linearization of the map, we Taylor-expand the nonlinear term of

equation (2. 18), and keep the first four terms to get

—K _ a
r "If: (f0 + y; - y.-.) = co + c. (y.- - y.-.)+ c,(y,. - y,._,)2 + c,(y, — y,._,)3 (2.22)

where
c = -——K"0W;
m9
Co = Cfo";
c, = cqfo‘“; (2.23)

c2 = ca(a - 1) fog—2;

c, = ca(a -1)(a — 2) foa"3

29

Substituting equation (2.22) into the equation (2.18), and rearranging it, yields

yi __ yi-l 0
[vii— ”[vr-r ]+[C0 +C2()’,- - y,_,)2 +C3(y‘. — yH)3] (2.24)

where the matrix B is defined as

_ Arr A12
B ._ (2.25)
A21 +C1(1—Ar|) A22 ‘CrArz

where A1), A12 , A21 , A22 are elements of Matrix A.

We make a coordinate transformation to shift the fixed point to the origin:

WW)
v, v, v

Substituting into equation (2.24), we get a new equation,

.Zi = 314+ ~ ~ 20 ~ ~ 3 (2.27)
V: Vr—i 02(Yr—Yr4) +C3(}’r‘)’r-r)

whose equilibrium is (0,0). For simplification, we drop the “~” in equation (2.27)and

y.- = y.-. + 20 3 (2.28)
V: ”H c2()’r"’)’r-i) +03(Yr'Yi—r)

This is the equation we will deal with in the rest of this work. The stability of the

write it as,

fixed point of equation (2.28) is determined by the eigenvalues of matrix B. Particularly,
if either of the eigenvalues of B lies outside of the unit circle in the complex plane (i.e.,
has magnitude greater than 1), then the magnitude of any perturbation about the fixed
points will grow without bound on further iteration of the map; the system is unstable.
Conversely, if both eigenvalues lie within the unit circle, all perturbations about the fixed

point will decay to zero if given enough iterations; the system is stable (Devaney (1987) ,

30

Moon (1992)). When one or both of the eigenvalues of B lies on the unit circle, the
stability is borderline, and parameters that cause this to occur lie on the stability boundary
for the system.

Note the eigenvalues of matrix B are functions of chip width w. So for a given
system, if all the other parameters are fixed, there must be a limit chip width, denoted as

wnm. When w < wlim , the eigenvalues lie in the unit circle and the cutting is stable; when

w > wlim , one of the eigenvalues or both eigenvalues lie outside of the unit circle and the

cutting becomes unstable.

The linear stability has been analyzed by Davis et al. (1999a), and B adopted
through the rest of this section is for the case of digressive cutting forces.
The eigenvalues of B are calculated from the characteristic equation and can be

written as:

22 - Tr(B)/I + |B| = o (2.29)
where 2. are the eigenvalues of B, and Tr(B) and |B| are the trace and determinant of B,

respectively. It can be shown that Tr(B) and |B| can be expressed relatively simply in

terms of the components of the matrix A and the width of cut as follows:

Tr(B) = Tr(A) — clA12

|B| = W _ Cl A12 (2.30)

Without any further analysis, a useful conclusion can be drawn from equation
(2.30): for values of the parameters such that A12 = 0 the characteristic equation for the

system reduces to the equation for a damped simple harmonic oscillator, and the system

31

is stable. Examining equation (2.10), and defining 27m, = (0d , where fd is the damped
frequency in HZ, and (ad is the damped frequency in radius per second, we see that

Al2 = 0 when the following condition is satisfied:

2fd(l—p)

n

N§2= n=1, 2, 3,--- (2.31)

Recall that N is the number of flutes for the milling problem and is equal to l for

interrupted turning. Thus, for an interrupted cutting operation with small p the most

2fd _f_;

stable engagement frequencies are approximately 2 fd, fd,— 3 , Alternate

(n = 2, 4, 6, --) stable speeds fd— , f—"— , 215d— ,--- are recognized as approximately the same

as the optimally stable speeds for a full immersion machining operation. However, the

2f.: 2f__.._

frequencies 2 fd, 3

-- (n =1, 3, 5, ---), are new stable speeds not predicted for

full immersion cutting. These new stable speeds. were tested and experimentally and
numerically verified by the theory of partial immersion machining presented in Davies et
al. (1999b).

An exact stability criterion can be derived from further analysis of equation
(2.29). There are three possible ways for the system to become unstable: (1) a real-valued
eigenvalue passes through positive one travelling to the right on the real-axis; (2) a real-
valued eigenvalue passes through negative one traveling to the left on the real-axis; and
(3) a complex-conjugate eigenvalue pair passes outside the unit circle in the complex
plane. Stability criteria can be determined simply by examining the implications of these

conditions on equation (2.29). Each case will now be considered separately.

32

/\ A Im /\ Irn

/ \> \>
k R. J... JR.

(1)/i=+l (2)/1=—1 (3)|}\.|=l

 

 

 

Figure 2.5 The ways the eigenvalues travel through the unit circle

Case 1: xi = +1
Setting /l = +1 in equation (2.29) produces the following equation.

1— Tr(A) + M] = o (2.32)
This equation is independent of the width-of-cut w, and cannot be satisfied for non-trivial

values of the parameters. Thus, 2 = +1 is not a route to instability for this system.

Case 2: xi =-l

-2§w.(I—p)
Setting l=—1 in equation (2.29) and noting that |A|=e 9 and that

-€w.(1-p) __
Tr(A)=2e ‘2 cos(—a-)—"£:-2——p—)-

), the following expression relating wum and .{2 can be

derived.

1— p l— p
1 med .cosh({a)n —S-2——) + cos(a)d —Q-)

 

_ , (2.33)
a-l _
a f0 pK, sin (‘04 1 p)

33

The numerator of equation (2.33) is always positive. Thus, requiring w“m to be positive,

implies that the sin term in the denominator must be positive. This occurs for values of .Q

in intervals defined as follows:

6040- p) < 9 < 0120-10) k =1,2,3,... (2,34)
(2k +1)7r 2k7r
For all other values of .Q, w“m is infinite or negative; these cases are non-physical and are

disregarded.

Case 3: AZ- : 1
This case can be related aszl/T =1, where I is the complex conjugate of it. This
condition implies that |B| =1. Applying this condition, another expression relating wlim to

.Qcan be derived:
. h C l—p
1 _-2QOd .srn ( a)" Q )

— a—l _
(#0 pK, sin(wd ——1 Q )

 

(2.35)

The numerator of equation of (2.35) is always negative implying that, to obtain positive

values of w,inn , the values of [2 must be in the following intervals:

coda-mayhem

k = 1,2,3,--- (2.36)
2k7z (2k — I)”

At speeds where sin[-&(IQ-—’o)] = 0 , equations (2.33) and (2.35) imply infinite

stability. This is consistent with the observation made above that the characteristic

equation reduces to that of the damped simple harmonic oscillator at these speeds.

34

Taken together equation (2.33) and (2.35) provide expressions for those
combinations of w and [2 that produce borderline stability in the system. Plots of these
parameter values as demonstrated in Figure 2.6 are analogous to the stability lobe
diagrams for uninterrupted machining.

For example, consider the interrupted turning of an aluminum workpiece as
shown in Figure 2.1. For the purpose of illustration, let us assume the following
reasonable parameter values for the modal stiffness, mass and damping of the cutting
tool: k=1 MN/m, m=0.05 kg, c=8.9 N—s/m, fo=0.1mm/revolution. This gives a damped
natural frequency of 711 Hz and a damping ratio 2%. In addition, assuming a reasonable

value for K, of 500 N/mmz, a =0.41 and p=2%, the stability lobe diagram can be

calculated from equations (2.33) and (2.35).

 

 

icV2 2id/3 id 21d
I r l l ‘I l I T fi/
l '_
, /
l / _
A I] / _
E I
g l / _
é ,' ’
._ / ‘
3 , ,
1% l / "
O , I
/ . / “
. l ’
I . / ‘
, L
| l l 1 l l l

 

 

 

 

 

 

 

 

30 4o 50 60 70 80 90 100 110
Spinde Speed (KRPM)

Figure 2.6 Stability lobes diagram of interrupted turning

35

Figure 2.6 shows the stability lobes of the interrupted cutting system over a wide
range of spindle speeds (some obviously not attainable). The lobes determined by

equation (2.33) are represented with dotted line and the lobes determined by equation

2
(2.35) are represented with a solid line. The asymptotes at ——f’-’— are clearly visible as
n

large narrow regions of stability. Note that in contrast to traditional regenerative chatter
stability where the highest speed stable region is at a spindle speed equal to dominant
frequency of the system, the last optimally stable region for interrupted machining is at
twice that frequency.

Traditional regenerative chatter theory predicts that for a single degree of freedom

system the most stable speeds are at integer fractions of the natural frequency of the

i" , k = l,2,3,---). The new theory predicts a set of stable speeds based on

 

system (i.e.

-. For small

fractions of the damped natural frequency defined by 35—9549, k =1,2,3,--

damping and small p , a subset of these stable speeds are approximately the same as

those predicted by the traditional theory. The others are not. A plausible physical
explanation of these new speeds is as follows. In partial immersion machining, the
location of the surface is determined by the location of the tool when it makes the cut.
Stable machining is most likely to occur when the tool returns to the same location at the

beginning of each cut. This can occur in two ways: (1) the period of the spindle is an

integer multiple of the period of the tool —1— indicating stable spindle speeds of
d

36

f4

7, k = l,2,3,---; or (2) the period of the spindle is an odd integer multiple of one half of

2
I]? , k =1,3,5,-~. Taken together

 

 

the period of the tool 21 indicating stable speeds of

d
these stable speeds are approximately those given by equation (2.33) and (2.35). This is
an appealing and easily remembered “rule of thumb”.

The type of instability , or more formally the form of the bifurcation that leads to
instability is different for the two types of lobes. On the lobes shown by the solid lines,
the instability results forma pair of complex eigenvalues passing through the unit circle in
the complex plane. In the dynamics literature, this is known as Hopf bifurcation. This is
same type of bifurcation that occurs in traditional regenerative chatter theory. On the
lobes shown by the dotted lines, the bifurcation occurs as a real eigenvalue passes
through minus one. This type of bifurcation known as a period-doubling bifurcation has
not been previously been observed in machining. This implies that the post chatter
behavior will be different depending on the spindle speed. Experimentally it was
observed as a difference in the sound of the chatter by Davies et al. (1999b).

While this type of interrupted turning operation is relatively rare, interrupted
milling operations are common. Davies et al. (1999b) demonstrated in a simple

experiment that the theory presented applies to practical milling operations as well.

37

Section 2.4: Local Bifurcations

To study the bifurcation of the map with one parameter changing, we assume that
all parameters, except the chip width, w, are fixed. We consider the period-doubling

bifurcation and Hopf bifurcation respectively.
Case 1: Period-doubling bifurcation

From Figure 2.6, we can see, those stability lobes act as the critical limits for both
the chip width and spindle speed. It is also called the parameter space in dynamics. If we
fix the spindle speed [2 at a certain value, the dynamics change with the chip width w,
which causes bifurcation. To make the problem simple, we choose the spindle speed .{2 to
lie approximately at the middle of each lobe. As such gives the lowest limit chip width

Wlim at each lobe for Hopf bifurcation (Tlusty (1986)).

Q]: = wda-p)

1 (k = 0, 1, 2, ~--) (2.37)
(2k + -2—)7r

2wd(1-p)

When k=0, :2, =
72

, corresponding to the last lobe, and the biggest lobe for the

period-doubling bifurcation at the right hand side of stability diagram. The parameter

traces are shown in Fiugre 2.7.

38

 

 

 

 

 

 

 

 

 

 

i i -' I' i ,

i i .I j ' 1'

r i l I. t '1'

i I. i I. : I”
.45 ; .- l i .
E i i l j . ’1
3 % 3 ! .I :
.9 I i I l 1"
J: I l i + I I
U | .' " .l. i II

I"! 'J \ I
>
[2» gut Spindle speed
Figure 2.7 Parameter traces for bifurcations
With the spindle speed [2 fixed at [215, matrix A becomes:
4 I f {a}, i )
(2k+—)n _ _
J _ 2 2 w a)
A, = e ' i 4;: {2’0” (2.38)
K wd wd J
The limit chip width where the bifurcation occurs becomes wk, where
2
m l p "’4 ~cosh( 5 (2k + ion) (2.39)

 

 

wk: . ' a-l
10K: (2k+_:_)7t (To Vl-gz

and consequently, the coefficient c) becomes

4 1
c =w ~cosh——— 2k+—72 2.40
. (7—1—45 2)) ( >
To use the suspension trick of center-manifold theory, we split c, in two parts as
(2.41)

Cr = Cu: +#

Considering My, — y,_,) as a nonlinear term, then equation (2.28) become:

39

y.- z B. y.-. + 0 2 3 (2.42)
Vi vi-l ”(Yr—yr-r)+cz(yr‘_yr'-I) +C3(yi-yi—I)

where

B = A” A” (2 43)
k A21+C1.k(1"A11) A22 —C1.kA12

Notice that the eigenvalues of 8;, are —1 and 1.2, (llzl <1), and cz, C3 are linear functions
of ,u.

To simplify equation (2.42), we use the transformation

(Ml

where T is composed of the eigenvectors of B1,. The equation (2.42) becomes

xi =J- xi—l + f(xi-r’ui-r’”) (245)
“r “H 8(xr-r’ui-r’fl) .

where J is the Jordan form of the matrix Bk:

J- _l 0 (246)
7 0 2, '

Writing the center-manifold in the form
u = h(x,,u) = 0le + azx/J + a3,“2 (2.47)
then the dynamics on the center-manifold are
x, = —Jr,._1 + f(x,_1,h(x,_l,p),p) (2.48)

Substitute equation (2.47) and (2.48) into (2.45), solve for al, a2, 33, we can obtain the

map on the center-manifold:

x l—-) —x+b,,ux+b2x2 +1),pr +b4p2x+b5x3 + 0(4) (2.49)
where b,, b2, b3, b4, and b5 are constants. 0(4) means the fourth order or higher in

variables x and ,u, which are infinitesimal around the origin.

The normal form of this map is:

x l—) —x+b“ux+b,x3 +0(4) (2.50)

The sign of €1- determines the types of the bifurcation and the stability of the period-two
l

orbit. When %1 > O, the bifurcation is subcritical, and the period-two orbit is unstable.
1

When b—' < 0 , the bifurcation is supercritical, and the period-two orbit is stable (Wiggins,
1

( 1990)).

The dominant form of the relation between b1, b, and the parameters like
§,wd ,a can not be obtained, we turn to use some examples to show the bifurcation.

We use the same example as in Figure 2.6, the interrupted turning operation with

the parameters m=0.05kg, k=IMN/m, c=8.9N-s/m, fo=0.1mm/revolution, and §=Z%,
f,.=71 1Hz, p=2%, a = 0.41, K,=500N/mm2. Fixing the cutting speed at 33.5 krpm, we

can get the normal form of the map:

x 1—> —x - 0000337215112: - 8.79211x10'6x3 + 0(4)

So the period-doubling bifurcation here is subcritical, and the period-two orbit is

unstable. The bifurcation diagram is shown in Figure 2.8.

41

 

10 I I fir I I I 7 I I

amplitude (mm)
N OD -> 01 O) \l on

 

 

o 1 1 1 1 1 1 1 11 1
0 1 2 3 4 5 6 7 8 9 10

chip width (mm)

 

Figure 2.8 Subcritical period-doubling bifurcation diagram.

Case 2: Hopf bifurcation
For the same reason as in the period-doubling bifurcation, we fix [2 at the

following values for each of the lobes (as mentioned above, they are corresponding to the

lowest chip width, (Tlusty, 1986):

$2, =34?”— k=1, 2, 3,--- (2.51)
2k - — 7r
( 2)
When k=l, £2, = god—gm , which is the second to the last lobe, the biggest lobe for the
72

Hopf bifurcation at the right hand side of stability diagram.

Correspondingly, we get other parameters as A, , wk ,cL, , B, , and similarly,

42

 

 

 

 

 

 

 

{run 1 )
_ 5 1 (211—4.? w "3'
1-
A, e 4 60:" {a}: (2.52)
‘04 wd )
m 1— p 2103 . g" 1
“’1 = - - a_ -srnh( (2k -—)7r) (2.53)
”Kr (2k-1):: 9‘0 1 41-42 2
2
-2 -'h( 4 (2k—1 ) (254
cu, — cod srn ([1—7 2)” . )
A A
31 = ” ‘2 (2.55)
A21 +C1.1(1_A11) A22 _CchArz
Y1 = 31 YM + 0 2 3 (2.56)
V1 Vr—r #(Y1-Y1-1)+C2(y1_y1-1) +cs()’r “Yr-1)
The eigenvalues of 8;, take the form
A, 2 008,6 :t isin ,6 (2.57)
Transforming the coordinates system as
[flat] (2.58)
v u
where
l i
T = (el ,e,{ ] (2.59)
1 —1
and e1, e; are eigenvectors of Bk, the equation (2.56) becomes
xi ___ ODS/3 " 51“ fl . xi—l + f(x1-1 ’ui-l’”) (2.60)
It. 8111 I3 90813 “1.1 8(X.-1.u.-1.fl)

where ,u is defined as in case 1.We can transform the above equation into polar

coordinates and get the normal form (Guckenheimer and Holmes, (1983)).

43

rl-> r+d,ur+ar3 +0(4)

2 (2.61)
61—9 19+c+br +0(3)

The signs of the coefficients a and d determine the direction and stability of the
bifurcation periodic orbits; c and b give asymptotic information on rotation numbers.

The invariant circle is expressed as:

 

{(r,0)ER+xS'|r= _fld} (2_62)

a

It is asymptotically stable for a<0 and unstable for a>0 (Wiggins, (1990)).

Since a and d are the most important for determining the local bifurcation and the
periodic orbits, using some shortcuts to compute the a and d directly will let us know the
characteristics of the bifurcation immediately. From the process of normalization, we

know,

(2.63)

         

where 1(a) is the eigenvalue of matrix B.
Rewriting the equation (2.60) as
x H cos/3 win/9 , x + f(x.y.#) (2.64)
y smfi COS/3 y 8(x.y,#)
with the eigenvalues 2,2.— =cos(fl)isin(fl). The value of a can be obtained by

(Guckenheimer and Holmes, (1983)),

1- 2,12 —
—Re[(l——_ ’ 511520]—:—|§n|2-|502|2+RCM§21> (2.65)

where

€20 =30... —f,, +2g.)+i<g.. ~g, —2f.,)]

4:11 :3?ka +fyy)+i(gxx +gyy)]

1 (2.66)
602 =§[(f.. —f,, -2g.,>+i(g.. —-g,, +2f.,)]

1 .
€21=Ekfm +1“... +8.... +g,.,>+z<g... +g... -f.., —f.,,>]

All the variables above are evaluated at (x, y, ,u) = (0,0,0).
Similarly, the dominant form of the relation between d, a and the parameters like
4,01,, ,0 can not be obtained, we turn to use some examples to show the bifurcation.

We use the same example as in Figure 2.6, the interrupted turning operation with

the parameters: m=0.05kg, k=IMN/m, c=8.9N—s/m, fo=0.1mm/revolution, and so {=2%,
f,.=71 1H2, p=2%, a = 0.41, K=500N/mm2. Fix the cutting speed at 55.8krpm, we can get
the normal form of the map:

r 1—> r + 34.9892pr + 1.61396 x10'6 r3

So the Hopf bifurcation here is subcritical, and the period orbit is unstable. The

bifurcation diagram is shown in Figure 2.9.

45

 

SD
01
T
l

N
01
l

1

amplitude (mm)
M

 

 

 

1.5~ .
o.5~ ,
0 1 1 1 1 ‘1 1 1
o 0.5 1 1.5 2 2.5 3 3.5 4

chip width (mm)

Figure 2.9 Subcritical Hopf bifurcation diagram

Section 2.5: The Role of on

In this thesis, we assume the cutting force is a digressive function of chip

thickness, i.e.

F = Cd“ (2.67)
where F is the force, d is the chip thickness, and C is a constant determined by cutting
conditions.

This is an empirical relation between cutting forces and chip thickness. Pratt
(1999) showed that a = 0.41 in continuous cutting by the experiments. This makes sense

since when chip thickness d is very small, cutting force F increases quickly with the

46

increase of d; but when d becomes bigger, F increases more and more slowly with the
increase of d. This is a material property. The relation between F and d can not be linear
in real cutting. Actually the value of a has important role on stability and it determines

the subcritical/supercritical bifurcation.

We rewrite the equation (2.28) as

Y1“ = Yi-l + 0 2 3 (2.68)
Vi vi—l C105" Yi-1)+Cz(y1' ' yi-l) +C3()’i ‘ yi-l)

where cI , c2, c3 are functions of a

To investigate the role of a on the bifurcations, fixed all parameter values, so that
A, c are constants, (Note, c<0), and cl , c2 and c3 are only functions of a. From equation
(2.68), we can see that cl determines the linear part of the map, so it determines the
critical bifurcation point. In the stability chart, for a given spindle speed, cl determines
the limit chip width, W11".-

Reviewing the equation (2.33) and (2.35), with all parameter values fixed, the

value of wlim is only function of a. When a = l,

l—p l—p
h— a) —— + a) —

 

 

w _ med
lim — ' _
pK' sin(w. 1723-) (2.69)
M<Q<M k=l,2,3,-~
(2k +1)” Zkfl'
. h 4. l— ,0
W _ - Zmed .Sm ( a)" $2 ) (2 70)
km pK; l— .

 

sin(wd

g2)

47

w.<1—p)<9<w.(1—p)

k = l,2,3,---
2kfl’ (2k — D7!

which are exactly the results of the linear model investigated by Davis (1999a). For

O<a<l,theterm

 

(H is always smaller than 1, so wlim from the nonlinear model is
0

always smaller than the corresponding one from the linear model. In figure 2.10, the
stability charts of two cases: a = 0.41 and a = l, are compared. It can be seen that the
stability charts from the nonlinear model are lower than those from linear model. This
partly explains why the threshhold of stability predicted by a linear model can not be set
up in actual experiments. The stability charts predicted by a nonlinear model is closer to

experimentally determined stability charts than the ones predicted by a linear model.

 

 

Chip Width (mm)

 

   

 

 

 

 

 

 

 

 

i l l l t l L l
20 30 40 50 60 70 80 90 100 110
Spindle Speed (KRPM)

 

Figure 2.10 The comparison of stability charts from the nonlinear model and the
linear model. ( ---- linear model, —. nonlinear model )

48

To see the role of a on subcritical/supercritical bifurcations, let us review the
process of simplification and normalization of the map (2.28).

For period-doubling bifurcation, the normal form is:
x i—) -x+b,,wc+b,x3 +0(4)
Comparing it with equation (2.68), and reviewing the process in Section 2.4, we
can imply that bl is totally determined by C1 , i.e.
bl = f(cl) (2.71)
Actually, with all parameter values fixed,
b. = not“) (2.72)
b,is determined by c2 and c3, i.e.
b, = f(c2,c3) (2.73)
Note, Qfom' > 0 for all a > 0 , so bl may not change the sign when (1 goes from
[0,1] to [1,00]. But c2 and c3 may change their signs and therefore b, may change its
sign, so the bifurcation may change from subcritical to supercritical when a goes from
[0,1] to [1,00]. Because the dominant form of the relation between b, and a can not be

obtained, we demonstrate several cases of the value of a.

For example, the interrupted turning operation with the parameters, m=0.05kg,

k=1MN/m, c=8.9N-s/m, fo=0.lmm/revolution, and therefore §=2%, wn=7lle, p=2%,
K=500Nlmm2. Fix cutting speed at 33.5krpm, try the following values of a, we get the

corresponding normal forms.

49

a__.0_11, —x— 0.000337215ux~ 0.0000102382 x3

 

(1:041, —x- 0.000337215ux—8.79211xlO‘6X3
0:090, —x— 0.000337215ux—2.04524 x 10'6x3
00:100, —x— 0.000337215ux
(1:110, -x— 0.000337215ux+2.27179 x10‘5x3
(1:120, —x- 0.000337215ux+4.77012 x10‘5x3
0$2.00, —x — 0 . 000337215 u x + 0. 0000329126 x3
0.3
0.25 ~
0.2 E:\
0.15 - ‘
0.1 ~
0.05 -

 

 

 

 

Figure 2.11 Period-doubling bifurcation diagrams at different values of oz

50

Figure 2.11 showes the bifurcation diagrams at different values of 04 From the
diagrams, we can see that, when 0 < a <1 , the period-doubling bifurcation is subcritical,
and the period-two orbits are unstable, so the hysteresis will happen; when a >1, which
is not physically possible, the period-doubling bifurcation would be supercritical, and the
period-two orbits would be stable, and there would be no hysteresis. When a =1, the
system degrades to a linear model. The chatter amplitude tends to go to infinity, in
contradiction to “the finite amplitude instability” theory. This is why a linear model is not
sufficient to describe the system when bifurcations occur.

For Hopf bifurcations, the role of dis similar. The normal form is

r l—> r+d,ur+ar3 +0(4)
where d is totally determined by C1 , i.e.
d = f (C1) (2.74)
Actually, with all parameter values fixed,
d = f(a f. “) (2.75)
a is determined by c2 and c3 , i.e.
a = f ((:2 ,c3) (2.76)
Similarly, d may not change the sign when 0' goes from [0,1] to [1,00], but a may

change its sign, so the bifurcation may change from subcritical to supercritical. Because
the dominant form of the relation between a and acan not obtained, we try several cases
of the value of a. We use same example of the interrupted turning operation. Fixing the

cutting speed at 55.8krpm, applying the following values of a, we get the corresponding

normal forms.

51

0t=0.11,

a=0.4l,

0t=0.99,

0L=l .0,

R+2

R+1

R+2

R+O.

.79039 x 10‘6 R3 + 18.7302 11R,

.61396 x 10‘5 R3 + 34 .9892 1112,

-73

.08262x10 R + 24.8538uR,

R3 + 21.9356uR

.81613 x 10'7 R3 + 19 .1665 uR’

—73

.36577x10 R + 16.6085uR’

.16911 x 10‘7 R3 + 4.38713 11R,

R=2.59083 x/ —u
R=4.65608 \/ -u

R: 10 .9242 \/ —u

R: 10.273 \III
R=7.02463 JG

R=2.66673 x/E

 

03

0255

02—

0.15L

01*

005

I

 

 

 

 

Figure 2.12 Hopf bifurcation diagrams at different values of on

52

Figure 2.12 showes the Hopf bifurcation diagrams at different values of a. We
can see that, when 0 < 0 <1, the Hopf bifurcation is subcritical, and the tori are unstable,
so hysteresis will happen; when 0 >1, which is not physically possible, the Hopf
bifurcation would be supercritical, and the tori would be stable, and there would be no
hysteresis. When a =1, the system degrades to a linear model. The chatter amplitude
tends to go to infinite, in contradiction to “the finite amplitude instability” theory. Again

the linear model is not sufficient to describe the system when bifurcations occur.

Section 2.6: Post-Bifurcation Orbits

2.6.1 Global Considerations

Local considerations are sufficient for understanding the consequences of a
supercritical bifurcation, which is a continuous bifurcation, while global considerations
are necessary to understand the consequences of a subcritical bifurcation, which is a
catastrophic bifurcation.

When a subcritical period-doubling bifurcation takes place, the branch of stable
periodic solutions that exists before the bifurcation (say w<w1,-,,,) continues as an unstable
branch of periodic solutions after the bifurcation (at w: wum). An unstable limit cycle
(period-two orbit) collides with the stable limit cycle (period-one orbit), and the two are
replaced with an unstable limit cycle of the lower period. The local state of the dynamical
system will be an attracting limit cycle for w< wum. However, for w> w,,-,,., the post-

bifurcation state of the system cannot be determined by local considerations alone; global

53

considerations are necessary. There are two possibilities. First, the system evolution may
be attracted to a remote solution, which is either bounded (fixed point, periodic solution,
quasiperiodic solution, chaos) or unbounded. Such a bifurcation is dangerous and is
typically accompanied by hysteresis. Second, the system response may explode into a
new attractor (possibly chaos) when w is slowly varied past wum, with the old attractor
being a proper subset of the new attractor. The transition from a periodic state to a
chaotic state following a subcritical period-doubling bifurcation has been termed
intermittent transition of type III to chaos by Pomeau and Manneville (1980).

When a Hopf bifurcation takes places, the branch of stable periodic solutions that
exists prior to the Hopf bifurcation continues as a branch of unstable periodic solutions
after the bifurcation. A quasiperiodic orbit collides with the period-one orbit, and the two
are replaced with an unstable limit cycle of the period-one. The local state of the
dynamical system will be an attracting limit cycle for w< wum. However, for w> w,,-,,,, the
post-bifurcation state of the system cannot be determined by local considerations alone;
global considerations are necessary. Similarly, there are two possibilities. First, the
system evolution may be attracted to a distant solution, which is either bounded (point,
periodic, quasiperiodic, or chaotic attractor) or unbounded. Second, the state of the
system may explode into a larger attractor, with the old attractor being a proper subset of
the new attractor. The transition from a periodic state to a chaotic state following a
subcritical Hopf bifurcation has been termed intermittent transition of type II to chaos

by Pomeau and Manneville (1980).

2.6.2 Case 1. Period-doubling Bifurcation

54

As explained in section 2.6.1, when the system undergoes the subcritical period-
doubling bifurcation as the chip width exceeds the threshold value wnm, the post-
bifurcation state of the system cannot be determined by local considerations alone.
Global considerations are necessary. What are the global considerations here specifically

in low immersion milling?

 

 

A
-8
2
'5‘
33 Period 2
‘5
.1:
U
S -N
1. i
5'0 ‘i >
w. Wlim Chip width

Figure 2.13 Bifurcation diagram for Hopf bifurcation in continuous cutting
(S-N: Saddle-Node bifurcation)

According to “finite amplitude instability” theory, the model of continuous
machining is based on the assumption that the tool will not leave the cut. The Hopf
bifurcation occurs after the chip width w becomes bigger than the threshold value an ,
and the “jump phenomenon” occurs. The amplitude of oscillation increases suddenly and
because of the nonlinearity of system, it will stabilize at a period-two orbit if the chip
thickness is big enough. This is shown in Figure 2.13 which is a Hopf bifurcation

diagram for continuous machining.

55

In low immersion milling, the amplitude “jump” in the subcritical period-
doubling bifurcation would cause the tool leaves the cut (run-out) immediately! This
makes sense because the immersion ratio is so small that the chip thickness is very small,
even a small increase in oscillation amplitude would cause the tool run-out.

What does it mean when the tool leaves cut? It means the tool does not engage in
cut during the subsequent revolution for interrupted turning, and the mill does not engage
in cut for the subsequent tooth for low immersion milling. What should be cut off by this
tooth will be left there and be cut off by the following tooth. Since the machine keeps
feeding in, the following tooth will cut off more than what it would otherwise cut. This
causes a bigger cutting force and results possibly in intensified oscillations. At the same
time, since the every other tooth does not engage in cut, the tool has more time to dampen
the oscillation, actually approximately twice the time as usual. It means the transition
time for the free vibration doubled. So there is a trend that the oscillation will be damped
more. There is a trade-off between these two effects—for example, the stable period—two
orbit may be created! This period-two orbit is characterized by “cut, no cut, cut, no
cut ...... ” for every other tooth in low immersion milling and every other revolution in
interrupted turning.

The simulations for period-two orbits in Chapter 3 will show this scenario. To
prove it theoretically, first we assume this period-two orbit exist, then we prove that it is
stable.

Basically this period-two orbit is the same as to the period-one orbit except that
the transition time (for free vibration) doubled.

We can write the similar map as:

56

Y" 1 y l 0
1+ = [4(2) '_ + K10 2.67
[1). J [V’l]_ _mpw (zfo + yi+l— yi—l) ( )

 

 

 

5w" sin(a)dr) + cos(wdz') —Lsin(wdr)
A”) ___ e-Ccm wd 2 (0" (2.68)
- . (U .
" sm(a)dz') cos(a)dz')— " srn(a)dr)
d wd
where
r = (2 — p)T = £3 (2.69)
82
The fixed point of map (2.67) corresponds to a period-two orbit on the Poincare
section.
[yi+l]_ [yi-l]_ [{J (270)
V141 vi—l v
Substituting (2.70) into equation (2.67), we get the fixed point for the period-two
orbits:

& =(I-A‘2’)"' _Kptg (271)
9 —m—Q_(2fo)a .

Expanding the nonlinear term in Map (2.67) into a Taylor series leads to

_K_._.3W_

m9 ___-(sz + 2’1"" 2’14)“ z C0 +C1(y1' — yi-l)+c2(yi — yi—l)2 +C3(y1' - yi-l)3 (2'72)

where CG, CI , c2 , c3 are Similar to those 1n the period-one map.

Combining the linear term c,(y,. — yH) with matrix Am in equation (2.67), and

transforming coordinates so that the fixed point of period-two orbit is relocated at origin,

we get the following map which is similar to the map for period-one orbit:

57

. 1 ._ O
yH-l : B(,_) yr l + 2 3 (273)
Vm vi—l 020'.“ ‘3’1—1) +C3(y1+1—)’1-1)

where

3‘” =[ Am“ Am” J (2.74)
Amy + c, (1 -— Am”) Amzz —c,A‘2’12

Whether the period-two orbit is stable or not is determined by the eigenvalues of
matrix B”). If the absolute value of the eigenvalues of the matrix B”) are in the unit
circle, the period-two orbit is stable; if not, it is unstable.

The eigenvalues of matrix B”) are functions of chip width w, so the stability of
the period-two orbit is determined by the chip width. There is a range of chip widths
(wum,,wum2) in which the period-two orbit is stable. When w< anu the period-two
orbit disappears, leaving the period-one orbit, i.e. the fixed point of the original map, as
the stable solution. When w> WW, another bifurcation occurs, and the way the
eigenvalues pass through the unit circle determines whether it is a period-doubling
bifurcation or Hopf bifurcation.

Furthermore, since the bifurcations from stable cutting are subcritical, the lower
limit of chip width for period-two orbit, wuml , must be smaller than the limit chip width
for the period-one orbit, wlim , which is defined in Section 2.3. i.e.,

wliml < wljm

This is the reason why hysteresis happens, this is why we call it catastrophe, and
this explains the “jump phenomenon”. The figure 2.14 shows the bifurcation diagrams of

this case. For the possible chaos, the chatter amplitude can reach any value in the shaded

region.

58

Possible chaos

Chatter amplitude

    

P

 

I
u..........—

4.-----

 

V

g
g.

Chip width

Figure 2.14 Bifurcation diagram for period-doubling bifurcation case

The value of wlimz can be determined by the eigenvalues of matrix 3(2) by using
the similar method in Section 2.3.

The determination of the value of wliml needs further global considerations. First,
let us assume that the chip thickness (depending on feed rate) is big enough that the tool
would never run out of cut during each engagement, even after the period-doubling
bifurcation, i.e., the cutting sequence is “cut, cut, cut, ...... ” for the assumed period-two
orbit. The bifurcation diagram of this case is similar to the one in Figure 2.13. Notice
that, when w < wliml , the period-one orbit is always stable, and there is a cut during every
pass (revolution), and no jump happens. Only after w> wIiml , it is possible for the

amplitude to jump, depending on the disturbance. After the jump, the chatter amplitude

59

will be stabilized at the real period—two orbit, which is a sequence of “cut, no cut, cut, no
cut ...... ”, before it reaches the assumed period-two orbit. Because the cut-no-cut
sequence has more damping time but less impulse than the cut-cut sequence, the
amplitude of cut-no-cut sequence must be smaller than the amplitude of cut-cut sequence.
There are several possible ways for the chatter amplitude to bifurcate to the real period-

two orbit as shown in Figure 2.15.

 

 

 

 

 

 

 

 

 

 

 

 

f0 f0 : ~
f.“\\ ’ ‘ J 1‘\ >
Wltm W Wlim W
(a) (b)
A A
Amp Amp
C-C . ,,,,,, C-C ,,,,,,,,
erml ’,”\’" ’,o’\”- C N
\: C-N Wlmrl 5: \
| \ I \
l: / \E \\\ /
f0 15 f0 4,
1 1 2’ 1 >
/ \ \
WC Wlim W Wlim W
(C) (61)

Figure 2.15 Possible ways that chatter amplitude stabilizes at the cut-no-cut
period-two orbit. (C-C: Cut-Cut period-two orbit; C-N: Cut-No-cut period-
two orbit.)

In Figure 2.15, case (a) is not feasible for low immersion machining because the
chip thickness is so small that any jump of the amplitude would cause the tool leave the
cut. It is almost impossible to get any stable cut-cut period—two orbit. Case ((1) is also not
very possible because it assume that the cut-no—cut period-two orbit only exists after

w> wnm, which is not a physically real case. Case (b) and case (c) are physically

possible. In case (c), when w < w < w , there should be some kind of periodic orbits

liml
which are not cut-no-cut period-two orbits. For example, it is possible to be a period-
three orbit, like “cut, cut, no cut, cut, cut, no cut ...... ”. But we will see in the
simulations, that these assumed periodic orbits do not exist for some typical low
immersion machining cases. In contrast, the cut-no-cut period-two orbits always exist

whenever the “jump” happens. So case (b) is the most physically possible case. In case
(b), the wliml can be determined by the following equation:

5*; - 5’1 = f0 (2.75)
where 5:2 is the fixed point of the period-two orbit, and 52, is the fixed point of the period-
one orbit. This can be explained as following, “as the chip width sweeps backward, the
cut-no-cut period-two orbit remains stable. But the amplitude will be smaller and smaller
until it reaches the surface of workpiece. After that, the period-two orbit is not of C-N

form, and vanishes, leaving the period-one orbit”. Substituting equation (2.21) and (2.71)

into equation (2.75), we can get,

0 O
_ _ fo
_ 4(2) 1 K pw _ _ A 1 K pw _—

V

where v is an uncertain velocity value.
From equation (2.76), we can get the analytic solution of ani as the function of
spindle speed [2, given other parameter values.

61

For example, we return to the interrupted turning operation with the parameters
m=0.05kg, k=1MN/m, c=8.9N-s/m, f0=0.1mm/revolution, and {=2%, a),.=71 1H2, p=2%,
a = 0.4land K,=500N/mm2. If we fix cutting speed at 33.5krpm, which is the case for
period doubling bifurcation in Section 2.4. we can also get the limit chip width for
period-one orbit,
Wlim = 13.4 (mm)

The lower limit chip width for the cut-no-cut period-two orbit is:
wliml = 9.9 (mm)

The upper limit chip width for the cut-no-cut period-two orbit is:
wlimz =80.7(mm)

When w = anz = 80.7(mm) , the eigenvalues are:

2 =-0.4837+O.8752i, I = -0.4837-0.8752i

I»: =I/TI

So the next bifurcation for the period-two orbit is a Hopf bifurcation. And this
Hopf bifurcation is similar to the Hopf bifurcation for the period-one orbit. After this

Hopf bifurcation occurs, the system dynamics becomes complicated, perhaps going to

chaos as shown in Figure 2.14.

2.6.3 Case 2. Hopf Bifurcation

Now let us consider the post bifurcation state of the system after the Hopf

bifurcation occurs for the period-one orbit.

62

Toms

Chatter amplitude

 

 

 

 

Wliml Wlim Chip width
Figure 2.16 Bifurcation diagram for Hopf bifurcation case

Similarly, when the Hopf bifurcation occurs, the cutting will leave the cut, but not
so regularly any more. The tool will engage in the cut for several passes, then leave the
cut for a couple of passes. The number of passes in cut against the number of passes out
of the cut is usually not determined. It depends on the certain value of the chip width and
the spindle speed. The state of system can not be represent by an analytic solution as
equations (2.67), (2.68) and (2.69), and there are no longer fixed points for period-two
orbits. Instead, there is a two-period quasiperiodic orbit (torus) because the newly
introduced frequency and the frequency of the periodic orbit that exists prior to the

bifurcation are incommensurate (two frequencies a), and (02 are said to be
. . 0),. . . . . . . .
incommensurate 1f ——rs an irrational number). This quasrperrodrc orbit can not be

(02

represented by an analytic solution, but can show up in the Poincare section of the period-

63

two orbits, which should be a closed circle. This will be discussed later in Chapter 3. The
bifurcation diagram can be shown in figure 2.16.

Again, because of the hysteresis, for backward sweep, there must be a limit chip
width, wliml that, when w < Wlirnl , the quasiperiodic orbit jumps to the period-one orbit.
To find this wnml, again, we assume the cut-cut period-two orbit exists if the chip
thickness is big enough. This is similar to the case of the continuous cutting shown in
Figure 2.13. For the cut-cut period-two orbit, at a critical chip width, we , a saddle-node
bifurcation occurs, and the lower branch of the cut-cut period-two orbit is unstable. But in
real machining, before the chatter amplitude stabilizes at the assumed cut-cut period-two

orbit, it jumps out of the cut, and the orbit becomes quasiperiodic for the Hopf bifurcation

case. The maximum amplitude of the quasiperiodic orbit must be smaller than the cut-cut

 

 

 

 

 

 

 

T A
Amp Amp
C-C ,,,,,, C-C '''''''
v”>" “liml ’,’>'" Q-P
vvliml '5‘ \|‘\
\ |\
2 \\ Q-P 1: ‘\
f" i 1 “1 , f” $1 ‘3 ,
\ \
Wu... W We Wu... W
(a) ( b)

Figure 2.17 Possible ways that chatter amplitude becomes quasiperiodic.
(C-C: Cut-Cut period-two orbit; Q-P: Quasi-Periodic orbit.)

period-two orbit. Similar to the period-doubling bifurcation case, there are two physically
possible ways for the system becomes quasiperiodic as shown in Figure 2.17. Case (a) is

more possible than case (b) as the implications of period-doubling bifurcation. But

because there are no period-two orbits after the Hopf bifurcations. We can not get the

analytic solution of wliml . We have to turn to simulations.

Because the range of limit chip width for the stable period-two orbit after the
period-doubling bifurcation and the range of limit chip width for the quasiperiodic orbit
after Hopf bifurcation are very big compared to the limit chip width for the period-one
orbit, and because the immersion ratio is so small in low immersion milling that it can be
assumed to be an impulse model, so further studies for bigger chip width are meaningless
for the analytic solutions of low immersion milling because the basic assumptions for the
impulse model no longer hold. Further studies can be carried out by simulations with the

method discussed in Chapter 3.

65

CHAPTER 3

SIMULATIONS AND EXPERIMENTS

Section 3.1 Simulations

The theoretical aspects of chatter can be well illustrated and understood by using
time domain simulation. The time domain simulation of machining operations is to be
done for a large number of revolutions at given speed, feed, chip width and other cutting
' parameters. This approach not only permits a good insight into the behavior of the
vibrating system, but also makes it possible to correctly take into account the basic
nonlinearity of the process and the effect of perturbations.

The nonlinearity of machining chatter is not only due to the fact that the cutting
force is nonlinear function of chip thickness and chip width, but also due to the fact that
when vibration grows larger, the tool jumps out of the cut for a part of vibrational period
and the cutting force disappears for this time instant.

Specifically, for interrupted turning and low immersion machining, time domain
simulation can accurately determine when the tool is in the cut, and when it is out of the

cut. That is, it can take into account the global considerations in determining the post-

bifurcation state of the system. So the simulation is a good tool to verify the analytic
solutions though it has the drawback that it is specific to a set of parameter values and

lacks the versatility and generality of the analytic solution.
3.1.1 Algorithm

Time domain simulation is often time-consuming. Here, to make the algorithm
simpler, to well coincide with the analytic model, keeping in mind the assumption that
the immersion ratio is so small that the machining operation can always be simplified into

an impact model, a simple algorithm can be implemented as following. An initial

condition, (y0,vo) , evolves through the time (1— p)T by using the equation (2.9), and

the state becomes (y..— ,vi' ). If the tool is going to engage in cut, the state (y,,v,.) results

from the impact model (equation (2.17)); if not, the state evolves through the time T by

free vibration. The simulation process is summarized as below:

(1) Initial conditions (yo .vo);

(2) [yi ] = {”4}, transition time 2 ;
Vi vi—l

(3) If y,- < y,._l +fo,then [y,]=[y:]’ set 1 =T ; goto (2);
v.

- 0
yr _ yi _ _ . .
else [v.]-[v7]+ _ Xi‘an/(f0 +Y1 _yHy], set 1 -(l p)T, goto (2),

67

3.1.2 Simulations of Stability Lobes

To obtain the stability chart by using time domain simulations is time consuming.
Analytical methods are preferred to time domain simulations if the analytical methods
are available. Usually, by using time domain simulations, only a certain number of
isolated points on the plane of stability chart can be obtained, each of which represents a
pair of spindle speed and chip width with other cutting parameters fixed. The stability of
these points can be determined by looking at the amplitude of disturbed vibrations. If the
amplitude is damping away and finally becomes zero, the point is stable, otherwise it is
unstable.

Setting a certain number of pairs of spindle speed and chip width with other
parameters fixed, and implementing the simulations using the algorithm above, we can
get a set of the unstable points on the plane of the stability chart. The comparison of the
analytic solution and the results of time domain simulations are shown in Figure 3.1.
From the figure, we can see that the results of the time domain simulations are coincide
well with the analytical solutions. The stability lobes of the Hopf bifurcation and the
period-doubling bifurcation are verified. Notice that, there are some regions where the
stability is conditionally held. It means, they are stable if the disturbance is small, but

unstable if the disturbance is big. This will be explained later.

3.1.3 Post Bifurcation Orbits

68

In chapter 2, we have discussed global bifurcations. After the subcritical period-
doubling bifurcation occurs, the tool leaves the cut and a stable period-two orbit is
created. The tool will engage in cut with every other tooth (low immersion milling) or
every other revolution (interrupted turning). We also find analytical representation for the
period-two orbit. After the subcritical Hopf bifurcation occurs, the tool also leaves the cut
but not in a periodic way. Instead, a quasiperiodic orbit is created. This quasiperiodic

orbit does not have an analytical solution.

 

 

 

 

 

 

 

 

 

 

 

 

 

$0 GOG
oc . 0005
cc . 000—
00 = ooo~
e 1 7
g oc 000
.C
§ oc = ooo~
3
% O< = 000—
0
o< = 000«
OOOAAAOOA oooa
ooOOOOOOOAoooooooooo~
o I l L l l

 

 

1o 15 20 25 30 35 4o
Spindle Speed (KRPM)

Figure 3.1 Comparison of stability lobes obtained from analytic solutions and from
simulations (‘x’—unstable points, ‘A’ —conditionally stable points, ‘o’-—stable

points.)

We can use simulations to verify these post-bifurcation orbits.

69

Because our concern here is only the state of system when the tool engages in the
cut during a revolution or a pass, while the system is otherwise just in free vibration, we
only record the state at the engagement time for every revolution or pass. We call these
records time series if they are plotted versus time. With the time series and their
Poincare sections, we can see the stable period-two orbit and the quasiperiodic orbit.

We return to the same example in chapter 2, the interrupted turning operation with
the parameters: m=0.05kg, fo=0.lmm/revolution, §=2%, (1).,=71le, p=2%, a = 0.41,
and K=500Nlmm2.

Figure 3.2 shows case 1: the stable periodic orbit after the period-doubling

bifurcation, where Q = 33.5krpm, w = 0.0174m, (wlim = 0.0134m).

Figure 3.3 shows case 2: the quasiperiodic orbit after the Hopf bifurcaiton, where
Q = 55.8krpm, w = 0.0046m, (wum = 0.0042m).

In Figure 3.2(a), the peak values of the oscillations represent those passes within
the cut; the minimum values represent the passes without a cut. These time series
correctly describe the characteristics of post bifurcation state of the system:

cut, not cut, cut, no cut ......

We use the Poincare section to show the characteristics of the orbits. That is, we
record the state of the system every revolution or pass at the engaging time. Figure 3.2(b)
and Figure 3.3(b) are the Poincare sections corresponding to each case respectively. In
figure 3.2(b), only two points can be obtained. One of them is the fixed point for the
stable period-two orbit which can also be calculated by using analytic method.

Substituting the parameter values used in the simulation into equation (2.70), we get

70

 

 

(a) Time series

 

 

3 . 1 r
-02 .......... s ..........
2 I I
-0.4 .......... g .......... g ......... 1
75‘, 1 » > o 6 __________ : _____ 7 - _ _ _: __________
E 3: ' . l l
0 8 I I
U
‘5 '5 I __________ I __________
E. 0 g > 50.8 ---------- I :
'U l |
-1 .......... I .......... I ......... J
-1 - : E -
-1.2 —————————— I —————————— I ——————————
-2 . 1 - . 4.4 I L
0 10 20 30 40 50 -1 0 1 2
time displacement
Figure 3.2 Stable period-two orbit after the subcitical period-doubling
bifurcation
(a) Time series (b) Poincare section
2.5 I V 0.8 l g l
2 - 05 .................. ”fit?! ....................
Q E Q.
15 ‘ [14 - ....... i. ............ ....... 3“... ......... _
1 t f = I ;.
.- 02.‘. ........... :9 ...........
E 0.5 - >‘ 3 E f ’6
E 3; 0 ...... ‘2 ........... :N: ........
8 U 8 1 :
2 E 02: ........... an‘ .......
8 -0 5 > ' ‘ t . : E :
:5 ' _04 ........ \ ........... .3 ........
-1 ' $2 3 8
.1.5- _06-...........’. .......... ........ ...;." ....... ..
_2 -0.8 .............. ”“oowx”. ............
-25 4 - - - -1 3 i
20 40 60 BO 100 -2 -1 0 1 2
time displacement

 

 

 

 

(b) Poincare section

 

 

 

 

 

 

 

 

Figure 3.3 Quasiperiodic orbit after the subcitical Hopf bifurcation

71

[9]_[ 1.7524 J
13 — 1.1048
which is exactly the lower point in figure 3.2(b).

In figure 3.3(a), the time series of the quasiperiodic orbit is shown, which can be
depicted in the Poincare section of Figure 3.3(b), in which a closed circle is obtained. To
show which passes engage in cut and which passes do not, the tool displacement is

plotted versus time along with the instantaneous workpiece surface as in Figure 3.4. the

displacements which coincide with the surface represent a cut, and those below the

surface are out of the cut.

 

OD #
T

displacements

 

"."surface of workpiece -
"-"orbits of tool
.4

 

.1

t I 1 l r

0 10 20 30 40 50 60 7o 80 90 100
time

Figure 3.4 Comparison of in-cut passes and out-of-cut passes for
quasiperiodic orbits.

72

3.1.4 Bifurcation Diagram

Using time domain simulations, it is possible to obtain the whole bifurcation
diagram from a stable fixed point to periodic orbits, or quasiperiodic orbits, and even to
chaos if the algorithm is accurate and simulation time is sufficient. But since here we are
concerned the cases around the limit chip width where the local bifurcations occur, we
just simulate a certain number of cases near the bifurcation points.

To simulate the bifurcation diagram, we fix the spindle speed at a certain speed,
change the chip width little by little, and record the amplitudes of each case. The

bifurcation diagrams of simulations are shown in Figure 3.4 and Figure 3.5

From Figure 3.4, we can see that when w< wnm, the period-one orbit is stable.
When w > wnm, the subcritical period—doubling bifurcation occurs, and the amplitude
jumps from zero to a finite value. There is a range of chip width (wliml ’wlimZ) for which
the period-two orbits are stable. When wliml < w < wum, the period-one orbit is

conditionally stable. That is, when the disturbance is small, it is stable; when the
disturbance is big, it becomes unstable and will stabilize on the period-two orbit. This is
called “finite amplitude instability”. In experiments, this will be shown as the forward
chip width and backward chip width have different jump points.

Figure 3.5 can explained similarly as Figure 3.4. The difference is that after the
subcritical Hopf bifurcation occurs, there is no stable period-two orbit. Instead, a two-
period quasiperiodic orbit is obtained. Notice that there are some “windows” for chip

width in which some periodic orbits might exist.

73

x10

 

5 T I T T I
4 ~ —smalll disturbance .
*—-large disturbance wlimz

amplitude(m)

 

 

 

 

0 0.005 0.01 0.015 0.02 0.025 0.03
width(m)

Figure 3.5 Simulations of subcritical period-doubling bifurcation diagram

x10.

 

amplitude

 

 

 

 

 

U 1 2 3 4 5 5 7 8 9
x 10'3
Figure 3.6 Simulations of the subcritical Hopf bifurcation diagram

74

Section 3.2: Experiments

In this paper, a nonlinear theory for predicting the stability of highly interrupted
cutting is presented. The theory is based upon modeling the cutting as a kicked harmonic
oscillator with delay which has been done by Davis et al. (1999a). On the contrary of the
traditional regenerative chatter theory, the most important prediction of the new theory,
from the practical perspective, is that the number of optimally stable speeds doubles as
the ratio of the time spent cutting to not cutting decreases. Davis et al. (1999b) did some
experiments to confirm this theoretical prediction.

The experiment below shows the new stable speeds with long-overhang end-
milling in Figure 3.6. The ratio of the length and diameter of the cutter is 9:1, and the
stability measurements were done for the tool for 25% and 5% immersion respectively.
Notice the new stable region of spindle speeds between 18 krpm and 20 krprn for 5%
immersion compared to 25% immersion.

Part of the stability charts from analytical prediction and from the experiments for
the 9:1 tool is compared in Figure 3.7. It can be seen that the experiment agrees with the

analytical prediction.

75

 

12 I I I I I I I I I

10 — O ~5% immersion ~

«11 —25% immersion

Chip Width (mm)
a:

 

 

 

4 _ _

>

. I
o 1 1 1 1 1 1 J 1 1

10 11 12 13 14 15 16 17 18 19 20

Spindle Speed (KRPM)

Figure 3.7 Comparison of stability charts from experiments of 5% immersion and
25% immersion end-milling. (Courtesy of M. A. Davis)

 

12

10

I

 

 

Chip Width (mm)
a:

 

 

 

 

 

 

 

o 1 1 l 1 L 1 L 1
10 11 12 13 14 15 16 17 18 19 20

Spindle Speed (KRPM)

Figure 3.8 Comparison of stability charts from analytical prediction and experiments
of 5% immersion end-milling. (Courtesy of M. A. Davis)

76

CHAPTER 4
CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
Section 4.1: Conclusions

1. An nonlinear model of interrupted machining is built. The machine tool structure
is represented by an equivalent single degree of freedom system with the cutting
forces by a digressive function (a) of chip thickness.

2. For highly interrupted machining, the system is simplified into an impact model
and an analytic solution is obtained.

3. The stability chart is analytically estimated. The number of stability lobes is
doubled.

4. Besides Hopf bifurcation, Period-doubling bifurcation is also found in interrupted
machining. Actually, the Hopf and period-doubling bifurcation occurs
alternatively with the increase of spindle speed at certain cutting conditions.

5. Both Hopf bifurcation and period-doubling bifurcation are subcritical as 0<01<1.

The effect of the value of 01 on the bifurcations is analyzed. When 0<01<1, the

77

6.

bifurcation is subcritical; when 01:1, the system becomes linear, and the chatter
amplitude would be infinite; when 01>1, the bifurcation is supercritical.

The nonlinearity is also introduced by the vibrating tool leaving the workpiece
frequently as the bifurcation occurs.

Global bifurcations are analyzed. Period-two orbits and quasiperiodic orbits are
predicted. Possible routes to chaos are discussed.

Numerical simulations verify the analytic results above. Hysteresis is found for
subcritical bifurcations. The stable period-two orbit and quasiperiodic orbit are

found and well agree with the analytic solution.

Section 4.2: Suggestions for future work

1.

Experiments are in badly need to verify the analytic and numerical results. Some
previous works (Shi, Tobias (1985), Nayfeh et al. (1997), etc.) have shown the
subcritical instability and the hysteresis by experiments in the continuous cutting.
Davies et al. (1999a, 1999b) have shown the doubling of the stability lobes with
the linear model by the support of experiments, which proves both Hopf
bifurcation and period-doubling bifurcation are possible in interrupted machining.
But no experiments have been oriented to the subcritical/supercritical bifurcation
in the interrupted machining.

Numerical simulations can be elaborated for low immersion milling to verify the

analytic prediction. The immersion ratio (p) varies after bifurcation occurs, so it is

better to use a shooting method to determine when the tool engages in cut, and

78

when it runs out of cut. Actually, further investigation on the low immersion
milling inevitably results in a two-degree-of-freedom model and analytic
solutions can no longer be obtained. Only numerical simulations are feasible.

3. Better mathematical model can be built with more considerations on the number
of degree of freedom and nonlinearities on stiffness, damping. And the cutting

force can be the nonlinear function of chip area or chip volume.

79

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